We investigate the dynamics of a system consisting of a simple, harmonic oscillator with small nonlinearity, small damping and small parametric forcing in the neighborhood of 2:1 resonance. We assume that the unforced system exhibits the birth of a stable limit cycle as the damping changes sign from positive to negative (a supercritical Hopf bifurcation). Using perturbation methods and numerical integration, we investigate the changes which occur in long-time behavior as the damping parameter is varied. We show that for large positive damping, the origin is stable, whereas for large negative damping a quasiperiodic behavior occurs. These two steady states are connected by a complicated series of bifurcations which occur as the damping is varied.
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ASME 2005 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference
September 24–28, 2005
Long Beach, California, USA
Conference Sponsors:
- Design Engineering Division and Computers and Information in Engineering Division
ISBN:
0-7918-4738-1
PROCEEDINGS PAPER
Parametric Resonance of Hopf Bifurcation Available to Purchase
Albert Barcilon,
Albert Barcilon
Florida State University, Tallahassee, FL
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Tina Morrison
Tina Morrison
Cornell University, Ithaca, NY
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Richard Rand
Cornell University, Ithaca, NY
Albert Barcilon
Florida State University, Tallahassee, FL
Tina Morrison
Cornell University, Ithaca, NY
Paper No:
DETC2005-84016, pp. 1557-1564; 8 pages
Published Online:
June 11, 2008
Citation
Rand, R, Barcilon, A, & Morrison, T. "Parametric Resonance of Hopf Bifurcation." Proceedings of the ASME 2005 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference. Volume 1: 20th Biennial Conference on Mechanical Vibration and Noise, Parts A, B, and C. Long Beach, California, USA. September 24–28, 2005. pp. 1557-1564. ASME. https://doi.org/10.1115/DETC2005-84016
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